The Dirac Field

The scalar field quantized spin-0. This page does spin-$\frac{1}{2}$: the free Dirac field $\psi (x)$, its quantization with anticommutators, and the appearance of antiparticles and the fermion propagator. The Dirac equation itself is derived along the historical route in QED/historical.md § 1.4; here it is taken as the classical field equation and quantized.

Conventions: $\hslash =c=1$, metric ${\eta }_{\mu \nu }=\mathrm{diag}(+1,-1,-1,-1)$, ${\omega }_{\mathbf{p}}=\sqrt{{\mathbf{p}}^2+m^2}$. Gamma matrices satisfy the Clifford algebra $\{{\gamma }^{\mu },{\gamma }^{\nu }\}=2{\eta }^{\mu \nu }$; $\bar{\psi }={\psi }^{†}{\gamma }^0$.

The classical Dirac field

The Dirac Lagrangian is

$$\mathcal{L}=\bar{\psi }(i{\gamma }^{\mu }{\partial }_{\mu }-m)\psi ,$$

a Lorentz scalar built from the four-component spinor ${\psi }_{\alpha }(x)$ (which carries the $(\frac{1}{2},0)\oplus (0,\frac{1}{2})$ representation of $SL(2,\mathbb{C})$; see preliminaries § universal cover). The Euler–Lagrange equation for $\bar{\psi }$ gives the Dirac equation

$$(i{\gamma }^{\mu }{\partial }_{\mu }-m)\psi =0.$$

Acting with $(i{\gamma }^{\nu }{\partial }_{\nu }+m)$ and using the Clifford algebra shows each component also satisfies Klein–Gordon, $(\square +m^2)\psi =0$, so plane-wave solutions live on the mass shell $p^2=m^2$.

Plane-wave spinors

The general classical solution is a superposition of positive- and negative-frequency plane waves with constant spinor coefficients $u^s(p)$, $v^s(p)$ ($s=1,2$ labels spin):

$$(/p-m)u^s(p)=0,(/p+m)v^s(p)=0,/p\equiv {\gamma }^{\mu }{p}_{\mu }.$$

They are normalized by ${\bar{u}}^r{u}^s=2m{\delta }^{rs}$, ${\bar{v}}^r{v}^s=-2m{\delta }^{rs}$, and satisfy the completeness (spin-sum) relations that recur in every cross-section calculation:

$$\sum _s{u}^s(p){\bar{u}}^s(p)=/p+m,\sum _s{v}^s(p){\bar{v}}^s(p)=/p-m.$$

Quantization requires anticommutators (Postulate — specializes QFT Postulate 9)

Naively one would impose commutators as for the scalar. This fails: the Dirac Hamiltonian obtained from the mode expansion would be unbounded below (the negative-frequency modes contribute $-{\omega }_{\mathbf{p}}$), so the vacuum would be unstable. The cure — forced, not chosen — is to quantize with anticommutators:

$$\boxed{\{{\psi }_{\alpha }(\mathbf{x}),{\psi }_{\beta }^{†}(\mathbf{y})\}={\delta }_{\alpha \beta }{\delta }^3(\mathbf{x}-\mathbf{y}),\{{\psi }_{\alpha },{\psi }_{\beta }\}=\{{\psi }_{\alpha }^{†},{\psi }_{\beta }^{†}\}=0.}$$

This is the concrete face of the spin–statistics theorem: a half-integer-spin field must be quantized as a fermion. The requirement is doubly enforced — anticommutators are needed both for a bounded Hamiltonian and for microcausality below. (The historic "Dirac sea" picture of filled negative-energy states is the old language for the same fact; the modern statement is simply: anticommutators, and there are no negative-energy states at all.)

Mode expansion and Fock space

The quantized field expands over the plane-wave spinors with ladder operators $a$ (particles) and $b$ (antiparticles):

$$\psi (x)=\int \frac{d^{3}p}{(2\pi {)}^3}\frac{1}{\sqrt{2{\omega }_{\mathbf{p}}}}\sum _s(a_{\mathbf{p}}^s{u}^s(p)e^{-ip\cdot x}+b_{\mathbf{p}}^{s†}{v}^s(p)e^{+ip\cdot x}),$$

with the non-vanishing anticommutators

$$\{a_{\mathbf{p}}^r,a_{\mathbf{q}}^{s†}\}=\{b_{\mathbf{p}}^r,b_{\mathbf{q}}^{s†}\}=(2\pi {)}^3{\delta }^{rs}{\delta }^3(\mathbf{p}-\mathbf{q}).$$

The normal-ordered Hamiltonian is then manifestly positive,

$$:H:=\int \frac{d^{3}p}{(2\pi {)}^3}{\omega }_{\mathbf{p}}\sum _s(a_{\mathbf{p}}^{s†}{a}_{\mathbf{p}}^s+b_{\mathbf{p}}^{s†}{b}_{\mathbf{p}}^s),$$

with both particles and antiparticles carrying positive energy ${\omega }_{\mathbf{p}}$ — the negative sign that plagued the naive attempt was absorbed by the anticommutator when moving $b{b}^{†}$ into normal order.

Pauli exclusion, automatically

Antisymmetry $\{a_{\mathbf{p}}^{s†},a_{\mathbf{p}}^{s†}\}=0$ gives $(a_{\mathbf{p}}^{s†}{)}^2=0$: no two fermions can occupy the same mode. The Pauli exclusion principle is thus a derived consequence of anticommutator quantization, not a separate postulate. Multi-fermion states $a_{{\mathbf{p}}_1}^{s_1†}\cdots a_{{\mathbf{p}}_n}^{s_n†}|0⟩$ are automatically antisymmetric (Slater-determinant structure), realizing Fermi–Dirac statistics.

Charge conjugation and antiparticles

The $U(1)$ Noether current of the Dirac field is $j^{\mu }=\bar{\psi }{\gamma }^{\mu }\psi$, with conserved charge

$$Q=\int \frac{d^{3}p}{(2\pi {)}^3}\sum _s(a_{\mathbf{p}}^{s†}{a}_{\mathbf{p}}^s-b_{\mathbf{p}}^{s†}{b}_{\mathbf{p}}^s)=N_a-N_b.$$

Particles and antiparticles carry opposite charge and equal mass — the electron and positron of QED. The operation exchanging them is charge conjugation $C$, treated with $P$ and $T$ on the CPT page.

Microcausality

The unequal-time anticommutator of the field is a c-number, and for spacelike separation the anticommutator does not vanish — but the physically required object, the (anti)commutator appropriate to observables, does. Concretely,

$$\{{\psi }_{\alpha }(x),{\bar{\psi }}_{\beta }(y)\}=(i/{\partial }_x+m{)}_{\alpha \beta }i\Delta (x-y),$$

with the same invariant $\Delta$ as for the scalar. For $(x-y{)}^2<0$ one finds that all physical (bosonic, fermion-bilinear) observables — currents, the Hamiltonian density, any $\bar{\psi }\Gamma \psi$ — commute at spacelike separation:

$$\boxed{[\bar{\psi }\Gamma \psi (x),\bar{\psi }{\Gamma }^{\prime }\psi (y)]=0\text{for}(x-y{)}^2<0.}$$

This is microcausality for fermions: observables are even in the fields, so pairs of anticommuting $\psi$'s recombine into commuting bilinears. Had one tried to quantize with commutators, the bilinears would not commute outside the light cone — the second, independent reason spin-$\frac{1}{2}$ must be fermionic.

The Dirac (fermion) propagator

The Feynman propagator is the time-ordered two-point function, with the crucial minus sign from fermionic time-ordering (see the time-ordering operator):

$$S_F(x-y)=⟨0|T\psi (x)\bar{\psi }(y)|0⟩,T\psi (x)\bar{\psi }(y)=\theta (x^0-y^0)\psi (x)\bar{\psi }(y)-\theta (y^0-x^0)\bar{\psi }(y)\psi (x).$$

In momentum space, using the spin-sum relations to collapse ${\sum }_{s}u\bar{u}=/p+m$,

$$\boxed{S_F(p)=\frac{i(/p+m)}{p^2-m^2+iϵ}=\frac{i}{/p-m+iϵ}.}$$

It is the Green's function of the Dirac operator, $(i/{\partial }_x-m)S_F(x-y)=i{\delta }^4(x-y)$, so — exactly as for the scalar — the external Dirac operators in the LSZ formula amputate external fermion legs. The overall sign and the numerator $/p+m$ are the fermion-line Feynman rule; closed fermion loops carry an extra $(-1)$ from the anticommuting reordering.

Summary: scalar vs. Dirac

Where this leads

• Applications of the Dirac equation (gyromagnetic ratio, fine structure): QED/dirac-applications.md.
• Coupling to the photon to build QED: The free vector field then QED from postulates.
• Grassmann path integral — the functional route to the same propagator and loop sign: Fermions & Grassmann integration.
• Discrete symmetries $C,P,T$ acting on $\psi$: C, P, T on fields.

References

• Peskin & Schroeder, An Introduction to Quantum Field Theory, Ch. 3.
• Weinberg, The Quantum Theory of Fields, Vol. 1, Ch. 5.5.
• Srednicki, Quantum Field Theory, Ch. 38–43.